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I. Introduction

In 1987 I decided to change my research profile.
Since 1982 I have been developing new hyperspace theories which
would allow for other "shapes" of internal space than were originally
allowed by Yang-Mills type generalizations of Kaluza-Klein theories.
Perhaps I need to explain first what I mean by the above, as it
does have a relation to my later research endeavors.

In 1921 Theodor Kaluza published a remarkable
paper "On the Unity Problem of Physics" [1] . He argued that if
we add fourth space dimension, then one force (gravity) in five-dimensional
(4 space + 1 time) splits into two different forces (gravity +
electromagnetism) as seen by us, who are blind to the extra dimension.

In 1926 Oskar Klein developed Kaluza's idea one
step further by showing that charge quantization can be explained
by assuming that extra dimension was "closed" into a circle. [EXPLAIN,
MAKE A DRAWING]. The one-dimensional rotation group acting on
this circle was related to "gauge transformation group" of the
electromagnetic field.

In 1952 C.N. Yang and R. Mills generalized electromagnetism
and introduced "non-Abelian gauge theories", where there is not
one, but many electromagnetic potentials, and where the "gauge
group" can be any group of not necessarily commuting matrices.
For instance, weak interactions are nowadays thought to be related
to the group of 2x2 complex unitary matrices of determinant one
- SU(2). Possible ways to generalization of Kaluza-Klein theory
with more than one extra dimension, so as to be applicable also
to other than electromagnetic gauge groups, were discussed by
several authors (J.M. Souriau (1962), B.S. de Witt (1964), J.M.
Rayski(1965), but a full geometrical and dynamical theory was
developed only by R. Kerner(1968), A. Trautman(1970), and Y.M.
Cho and P.G.O. Freund (1975).

Non-Abelian Kaluza-Klein theory (as it was called)
allowed for a unified description of gravity and of "gluon fields".
While photons are quanta of electromagnetic fields, and electromagnetic
field "glues" electrically charged particles together (thus attraction
of opposite charges), gluons are quanta of Yang-Mills fields that
glue together particles carrying charges other than electrical
(elementary particles are known to carry also leptonic charges,
hadronic charges etc.)

In the late 70-es we had a geometrical theory
that enabled us to derive interactions of all these charges from
one unifying force: gravity in hyperspace. The theory was really
neat mathematically, but it also had some drawbacks and lack of
naturality. The "internal space" in this theory had to have a
particular geometrical structure - it had to be a "group manifold".

To explain what "group manifold" means
let us consider the case of weak interactions. In electromagnetism
the internal symmetry group is O(2) - rotation group in a plane,
with one parameter - the rotation angle. For weak interaction
the internal symmetry group is O(3) - the rotation group of 3-dimensional
space, with three parameters (it looks like a 3-ball) that correspond
to independent rotations around each of three axes. Although the
group itself has three parameters, it can operate on a 2 dimensional
sphere. If so, then Occam's razor principle would suggest that
we should first try the internal space as a 2-sphere rather than
3-ball ("What can be done with fewer is done in vain with more").
That was, originally, the idea proposed by Souriau and de Witt,
but somehow, for years, nobody was able to convert it into equations.

In 1981 a paper by Edward Witten appeared that
generalized the "Kaluza-Klein Ansatz" (so it was called) to cover
the situations in which the internal space could be smaller than
the full group manifold, but the geometrical meaning of Witten's
paper was rather obscure. In 1982 another paper appeared, this
time by A. Salam and J. Strathdee, that developed Witten's ideas
further on, but without clarifying the underlying geometrical
principles. Formal operations on mathematical symbols could not
be converted into geometrical concepts.+